Solve 164=2a^2-36a+324

$$164 = 2 a ^ { 2 } - 36 a + 324$$

Solve for a

$$a=8$$

$$a=10$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$2a^{2}-36a+324=164$$

Subtract 164 from both sides.

$$2a^{2}-36a+324-164=0$$

Subtract 164 from 324 to get 160.

$$2a^{2}-36a+160=0$$

Divide both sides by 2.

$$a^{2}-18a+80=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as a^{2}+aa+ba+80. To find a and b, set up a system to be solved.

$$a+b=-18 ab=1\times 80=80$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 80.

$$-1,-80 -2,-40 -4,-20 -5,-16 -8,-10$$

Calculate the sum for each pair.

$$-1-80=-81 -2-40=-42 -4-20=-24 -5-16=-21 -8-10=-18$$

The solution is the pair that gives sum -18.

$$a=-10 b=-8$$

Rewrite a^{2}-18a+80 as \left(a^{2}-10a\right)+\left(-8a+80\right).

$$\left(a^{2}-10a\right)+\left(-8a+80\right)$$

Factor out a in the first and -8 in the second group.

$$a\left(a-10\right)-8\left(a-10\right)$$

Factor out common term a-10 by using distributive property.

$$\left(a-10\right)\left(a-8\right)$$

To find equation solutions, solve a-10=0 and a-8=0.

$$a=10 a=8$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$2a^{2}-36a+324=164$$

Subtract 164 from both sides.

$$2a^{2}-36a+324-164=0$$

Subtract 164 from 324 to get 160.

$$2a^{2}-36a+160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -36 for b, and 160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 2\times 160}}{2\times 2}$$

Square -36.

$$a=\frac{-\left(-36\right)±\sqrt{1296-4\times 2\times 160}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-\left(-36\right)±\sqrt{1296-8\times 160}}{2\times 2}$$

Multiply -8 times 160.

$$a=\frac{-\left(-36\right)±\sqrt{1296-1280}}{2\times 2}$$

Add 1296 to -1280.

$$a=\frac{-\left(-36\right)±\sqrt{16}}{2\times 2}$$

Take the square root of 16.

$$a=\frac{-\left(-36\right)±4}{2\times 2}$$

The opposite of -36 is 36.

$$a=\frac{36±4}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{36±4}{4}$$

Now solve the equation a=\frac{36±4}{4} when ± is plus. Add 36 to 4.

$$a=\frac{40}{4}$$

Divide 40 by 4.

$$a=10$$

Now solve the equation a=\frac{36±4}{4} when ± is minus. Subtract 4 from 36.

$$a=\frac{32}{4}$$

Divide 32 by 4.

$$a=8$$

The equation is now solved.

$$a=10 a=8$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$2a^{2}-36a+324=164$$

Subtract 324 from both sides.

$$2a^{2}-36a=164-324$$

Subtract 324 from 164 to get -160.

$$2a^{2}-36a=-160$$

Divide both sides by 2.

$$\frac{2a^{2}-36a}{2}=-\frac{160}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a^{2}+\left(-\frac{36}{2}\right)a=-\frac{160}{2}$$

Divide -36 by 2.

$$a^{2}-18a=-\frac{160}{2}$$

Divide -160 by 2.

$$a^{2}-18a=-80$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-18a+\left(-9\right)^{2}=-80+\left(-9\right)^{2}$$

Square -9.

$$a^{2}-18a+81=-80+81$$

Add -80 to 81.

$$a^{2}-18a+81=1$$

Factor a^{2}-18a+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-9\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-9\right)^{2}}=\sqrt{1}$$

Simplify.

$$a-9=1 a-9=-1$$

Add 9 to both sides of the equation.

$$a=10 a=8$$